Adverse selection and risk adjustment under imperfectcompetition

Normann Lorenz∗

March 22, 2014

Abstract

This paper analyzes the distortions of health insurers’ benefit packages due to ad-verse selection when there is imperfect competition. Usingthe (conditional and mixed)logit model for a theoretical analysis of market equilibriaunder adverse selection, forthe case of two risk types the following main results are derived: For intermediate lev-els of competition, the benefit packages of both risk types are distorted in the separat-ing equilibrium. If the level of competition decreases, thedistortion decreases for thelow risk type, but increases for the high risk type; in addition, the number of insurersoffering the benefit package for the high risk type decreases. If the level of competi-tion is low enough, a pooling equilibrium emerges, which generally differs from theWilson-equilibrium. It is shown that these results have important implications for riskadjustment: For intermediate levels of competition, risk adjustment can be ineffectiveor even decrease welfare if it is not reasonably precise.

JEL-classification: I 18.

Keywords: Adverse selection, discrete choice, imperfect competition.

∗Universitat Trier, Universitatsring 15, 54286 Trier, Germany; e-mail: lorenzn@uni-trier.de; phone: 0049-651-2012624. I thank Friedrich Breyer, Mathias Kifmann, Dominik Sachs and Esther Schuch and seminarparticipants at the HCHE (Hamburg) for helpful comments andsuggestions.

1 Introduction

Adverse selection has long been recognized as a potentiallyserious problem for insurancemarkets in general, and health insurance markets in particular.1 If individuals differ in theirexpected medical cost, but health insurers are not allowed to charge an individual-specificpremium, this creates incentives to distort the benefit package, so that the medical servicesoffered are attractive for some individuals, but not for others. Insurers who respond to theseincentives are said to perform indirect risk selection: they exploit adverse selection to in-fluence the risk structure of their insured. Several empirical studies have shown that thesedistortions exist and can be severe.2 Because many health insurance markets, especially inEurope, but also in the U.S., are characterized by communityrating, the number of individ-uals affected by these distortions is likely to be substantial.

Theoretical studies analyzing these distortions have usually considered the case of perfectcompetition, see, e.g., the highly influential study of Glazer and McGuire (2000). Healthinsurance markets may, however, not always be perfectly competitive. For the U.S., Dafny(2010) has demonstrated that in some markets, health insurers have a considerable degree ofmarket power.3 For the European context, Schut et al. (2003) and Tamm et al. (2007) haveshown that price elasticities of demand are low and that the number of individuals switchinginsurers is smaller than what would have to be expected in a perfectly competitive market.Some health insurance markets are rather imperfectly competitive.

This paper analyzes the interaction of these two phenomena –adverse selection and imper-fect competition – with a special focus on the distortions ofthe benefit packages offered.For health insurances markets, this interaction so far has only been examined for the fol-lowing two settings: In the first one, an arbitrary number of risk types is considered, buta pooling equilibrium is assumed, so that all insurers offerthe same contract, see Franket al. (2000). We relax this assumption and show that whethera separating or a poolingequilibrium emerges depends on the level of competition.

In the second setting, a separating equilibrium for the caseof two risk types is analyzed.Imperfect competition is captured with a Hotelling-model,where each insurer offers twocontracts (so that the incentive compatibility condition is satisfied); see, e.g., Olivella andVera-Hernandez (2007).4 This second class of models allows for sorting into different con-tracts, but also implies a strong asymmetry of demand responses which in some healthinsurance markets may not apply: Consider a group of individuals holding a contract froma particular insurer. A new contract, yielding slightly higher utility than the contract theycurrently hold, would attract all these individuals, if offered by the same insurer, but onlya small share of them, if offered by a different insurer. For some health insurance settings,this is a reasonable assumption and captures the behavior ofthe insured well. One exampleis a fee-for-service setting, where contracts differ mainly in the deductibles and coinsurance

1See Cutler and Zeckhauser (2000) and Breyer et al. (2011).2See Frank et al. (2000), Cao and McGuire (2003) and Ellis and McGuire (2007). For the distinction

between direct and indirect risk selection, see Breyer et al. (2011), p. 729; for an overview of risk selection, seevan de Ven and Ellis (2000).

3See also Cebul et al. (2011); for the Medigap market, see Maestas et al. (2009) and Starc (2013).4See also Biglaiser and Ma (2003), Jack (2006) and Bijlsma et al. (2011).

2

rates. Insured will easily switch to a different contract ofthe same insurer if it yields higherutility, but – being not perfectly informed about whether other insurers reimburse bills astimely and at the same level of generosity – may hesitate to switch to another insurer if thebenefit package itself is only slightly superior.

In some health insurance settings, however, insurers do notspecify reimbursement rates,but offer benefit packages of medical services, which may differ in the drug formularies, thephysician networks, the hospitals that can be attended or the disease management programsthat are implemented. In this case, from the perspective of an individual it will not makemuch of a difference, wether a new contract with, say a different physician network, isoffered by the same insurer or by a different insurer. For these health insurance settings, it isimportant to relax the assumption of a strong demand asymmetry, and we do so by analysinga model where each insurer offers only one contract, so that this asymmetry cannot occur.5

To keep the analysis simple, we consider the case of two risk types. If each insurer of-fers only one contract, but there are two risk types, a meaningful model that is supposedto also capture a separating equilibrium must comprise morethan two insurers. Therefore,a Hotelling-model is not appropriate. This is why we consider a discrete choice model,namely, the (conditional and mixed) logit model.6 The logit model has been extensivelyused in empirical analyzes of health insurance choice.7 In this paper we suggest that it isalso a very useful model for a theoretical analysis of marketequilibria under adverse se-lection when there is imperfect competition: It can captureany number of insurers, allowsto endogenize whether a separating or a pooling equilibriumemerges, and – by introduc-ing the concept of ‘indifference curve areas’ – has a graphical representation that providesan intuitive understanding of the economic forces driving the additional distortions underimperfect competition. It shows that of the two parameters that influence the level of compe-tition – the number of insurers and individuals’ responsiveness to differences in the benefitpackage – the latter is more important than the former. Finally, it captures the fact that someindividuals ‘make mistakes’ when choosing their health insurance contract, e.g., because ofinertia or information problems.8

For a very high level of competition, the discrete choice model replicates the results of amodel under perfect competition, where an efficient benefit package is offered for the highrisk typeH, and an inefficient one for the low risk typeL (Rothschild and Stiglitz 1976);see Figure 1, where we present – schematically – the distortions under perfect and imperfect

5One could also relax the assumption in a setting where each insurer offers more than one contracts. Thiswould complicate the model because each insurer would then have to take into account the consequences ofchanging one contract on the demand of its other contracts. If the number of insurers is large, this effect willbe small, but if the number of insurers is small, it may not be negligible. However, all the distortions we derivewould also occur in this setting.

6Olivella and Vera-Hernandez (2010) have analyzed a different extension of the Hotelling-model, the spokesmodel of Chen and Riordan (2007). They show that when each insurer can offer two contracts, a poolingequilibrium does not exist; also, an equilibrium where eachinsurer offers only one contract (but contracts differby insurer) does not exist either: At least one insurer offers both contracts so that the incentive compatibilityconstraint is satisfied. This implies the strong demand asymmetry which does not apply to the setting weanalyze.

7See, e.g., Feldman et al. (1989), Royalty and Solomon (1999), Harris et al. (2002), Keane (2004) andEricson and Starc (2012).

8See Handel and Kolstad (2013) and Sinaiko and Hirth (2011)) for empirical evidence.

3

competition.m∗ represents the efficient benefit package. Under perfect competition, whichis depicted at the beginning of the abscissa, we havemH = m∗ andmL < m∗.

perfectcompetition

high level ofcompetition

intermediate levelof competition

low level ofcompetition

(lower) level ofcompetition

m∗

mWI

mH

mL

level of benefitpackagem number of in-

surers offeringmH decreases,if competitiondecreases

Figure 1: Distortions of the benefit packages under perfect and imperfect competition.m∗

denotes the efficient level,mWI the Wilson-equilibrium.

With imperfect competition, we can distinguish three levels of competition: For high levelsof competition,mH is always at the efficient level, while, initially,mL decreases if the levelof competition goes down.

For intermediate levels of competition, bothmH andmL are distorted. If the level ofcompetition goes down,mH decreases andmL increases. In addition, the number of in-surers offering the benefit package for the high risk type decreases, until, for a low levelof competition, the pooling equilibrium emerges. For the pooling equilibrium, if the levelof competition goes down,m increases, so it coincides with the Wilson-equilibriummWI

only for one particular level of competition.9

These results show that imperfect competition has a direct effect on the distortions of thebenefit packages offered. The famous result of no distortionat the top clearly does not holdin general under imperfect competition: For intermediate levels of competition, both bene-fit packages are distorted in a separating equilibrium. Thisimplies that the most generousgenerous benefit package offered in a health insurance market may be a (severely) biasedindicator of the efficient level of medical services; this may contribute to explain why anumber of recent empirical studies have found the welfare losses caused by adverse selec-tion to be surprisingly small.10 However, these studies have estimated the welfare lossesdue to inefficient pricing of agivenset of benefit packages, but as explicitly stated by Einav,Finkelstein, and Levin (2010), the welfare losses due to an inefficient set of benefit pack-ages may be much larger, and we show, how these inefficienciesdepend on the level ofcompetition.

In the second part of this paper we show that the economic forces driving these results

9If mWI is very low, it may occur that all pooling equilibria of the discrete choice model are abovemWI .10See Einav, Finkelstein, and Cullen (2010), Bundorf et al. (2012) and Handel (2013).

4

have important implications for risk adjustment: For intermediate levels of competition, arisk adjustment scheme that is imprecise and only partiallycompensates insurers for thecost differences of different risk types may be ineffectiveor even increase distortions; atsuch levels of competition, risk adjustment only increaseswelfare if the cost differencesare reduced by a considerable amount. This contrasts with the case of either high or lowlevels of competition, where risk adjustment always increases welfare, even if transfersonly compensate cost differences to a small degree. With these results we add to the smallliterature that analyzes the negative side effects of risk adjustment.11

The remainder of this paper is organized in a way so that an intuitive understanding ofthe economic forces driving the results can be provided. We begin with the case that risktypes are observable in Section 2: We introduce the discretechoice model in Section 2.1and determine the equilibrium if there is one risk type in Section 2.2; we present an intu-itive graphical representation of this logit model in Section 2.3 by introducing the conceptof ‘indifference curve areas’ and briefly discuss the case oftwo observable risk types inSection 2.4.

We then analyze the case that risk types are unobservable – orthat risk types are observablebut community rating is imposed by regulation – in Section 3:12 We derive the separatingequilibrium in Section 3.1 and show, how it depends on the level of competition in Sec-tion 3.2. The pooling equilibrium is discussed in Section 3.3. We comment on the welfareeffects of different levels of competition for both the separating and the pooling equilibriumin Section 3.4 and provide an example in Section 3.5.

We consider the implications of our results for risk adjustment in Section 4. Using theexample introduced in Section 3.5, we first show that welfaremay decrease if a risk adjust-ment scheme becomes more precise (Section 4.1). We then explain why such a decreasecan only occur in a separating equilibrium (Section 4.2), but not in a pooling equilibrium(Section 4.3).

Finally, some of the assumptions of the model are discussed in Section 5, and Section 6concludes.

2 The discrete choice model

2.1 Basic model

We consider a setting as in Frank et al. (2000), where each individual may suffer fromS different illnesses. In case an illnesss is developed, utility changes byvs(ms), wherems is the medical services (measured in monetary terms) provided by the insurer;vs(ms)is increasing at a decreasing rate, i.e.v′s(ms) > 0 andv′′s (ms) < 0. The individual has

11See Brown et al. (2012) who show that for the U.S., the improvement of the risk adjustment schemeused for Medicaid has increased the incentive to enroll certain subgroups of individuals which are now evenmore ‘overpriced’ than before the reform; this increases health insurers’ wasteful expenditures to attract theseindividuals.

12Throughout the paper we will refer to this case by ‘unobservable risk type’; regarding the distortions weanalyze it is of course identical to the setting ‘observablerisk type under community rating’.

5

incomey and has to pay a premiumR. Utility is given by

u = y − R+

S∑

s=1

psvs(ms),

whereps is the probability for illnesss. The efficient level of medical services for eachillness is implicitly defined byv′s(m

∗s) = 1.

Insurers maximize profits by deciding which levels of medical services to offer and whichpremium to charge.13 It is straightforward to show that for all illnessess for which theprobabilityps is identical across individuals, insurers will offer the efficient level of medicalservices. Distortions only arise for those illnesses for which there is heterogeneity in risk.

To keep the model simple, we analyze the case where probabilities differ for only one of theillnesses, for which we assume two risk typesr = H,L, with pH > pL. Since insurers willoffer all the other medical services at the efficient level, we can skip these other illnesses(and incomey) to simplify the notation, and write utility as

u = prv(m)−R. (1)

We consider, however, the full model to be that in addition tom, insurers also offer theseother medical services (at the efficient level) and charge a premiumR that differs fromR

by the expected cost of these other illnesses. The distortions ofm that we describe shouldthus be considered to apply to a specific illness like, e.g., diabetes (or an illness category,like mental illnesses) rather than the overall level of medical services.14

There aren insurersj, each offering a contract{mj , Rj}. Within a discrete choice model,individuals’ utility as given by (1) is augmented by an insurer specific utility componentεij , that comprises all the influences on the choice of an insurerthat are independent ofmandR.15 The utility of an individuali (being of risk typer) when choosing an insurerjtherefore is

ui(mj , Rj) = prv(mj)−Rj + εij . (2)

We assumeεij to be i.i.d. extreme value, so that the logit model arises, but later show thatthe main results also hold for other distributional assumptions.

The level of competition is determined by the variance ofεij , Var(εij) = σ2 π2

6 :16 If σ islarge,εij assumes large positive and negative values, so the additional utility component isimportant; then competition with respect to different benefit packages is low: If an insurerraises its premium or lowers the level of medical services, only a few of its insured willswitch to another insurer. Because insurers are not close substitutes, each insurer has a

13We discuss the case that the premium is set by a regulator in Section 5.3.14Regarding the overall level of medical services it would certainly be more appropriate to assume a con-

tinuous distribution, but for an illness like diabetes or depression, the most important distinction is whether anindividual is chronically ill (pH = 1) or not (pL rather small).

15εij thus may capture, e.g., perceived friendliness of personnel, location, or, which insurer was recom-mended by family and friends, but it may also be unfounded andtherefore represent decision mistakes.

16Note that it is common to state the variance ofεij as a multiple ofπ2

6for the extreme value distribution,

see Train (2009, p. 24).

6

considerable degree of market power. If, on the other hand,σ is small,εij only has a smallinfluence on the decision of which insurer to choose, so competition is high. Withσ = 0, themodel encompasses the case of perfect competition.17 Forσ > 0, the level of competitionof course also increases in the total number of insurers,n.18

2.2 The Equilibrium with one risk type

Denote the utility component that does not depend onεij by19

V j = pv(mj)−Rj . (3)

Each individual chooses the insurer that offers the highestoverall level of utility, includingεij . Individual i will therefore choose insurerk if

V k + εik > V l + εil ∀ l 6= k.

For εij distributed i.i.d. extreme value with varianceVar(εij) = σ2 π2

6 , the probability ofindividual i choosing insurerk is given by20

Prob(i choosesk) =e

V k

σ

∑j e

V j

σ

. (4)

The mass of individuals is normalized to one, so that expression (4) also represents insurerk’s market share, which we denote byP k. Assuming profit maximization, the objective ofinsurerk is to maximizeπk = P kπk

i , whereπki = Rk − pmk denotes insurerk’s profit per

individual.

It will turn out much easier to derive the main results for thecase of unobservable risk typesif we reformulate the insurer’s objective in terms of{mj , V j} instead of{mj , Rj}. Graphi-cally, in anm-R-diagram, each insurerj chooses an indifference curveIV

jassociated with

the utility levelV j , and a level of medical servicesmj along this indifference curve.

Using (3) to substitute forRk, we therefore state insurerk’s objective as

maxmk,V k

πk = P kπki =

eV k

σ

∑j e

V j

σ

(pv(mk)− V k − pmk

). (5)

A convenient property of the market shareP k, which simplifies the derivation of the results,is that its derivative can be expressed in terms ofP k itself in a simple way:

∂P k

∂V k=

P k(1− P k)

σ. (6)

17Note that the parameterσ has a similar impact on the degree of market power insurers have as the para-metert, the transportation cost, in a Hotelling-model.

18We analyze the effect of different numbers of insurers, but do not endogenizen. However, this could easilybe done by assuming fixed costs of setting up a new health insurance.

19Because in this section we consider the case of only one risk type, we replacepr by p.20See Train (2009, p. 40).

7

Using (6), the FOCs for the insurer’s objective (5) are givenby

∂πk

∂mk= P k

[pv′(mk)− p

]= 0 (7)

∂πk

∂V k=

P k(1− P k)

σπki − P k = 0. (8)

Condition (7) requiresv′(mk) = 1, somk is chosen efficiently. Condition (8) shows the twocountervailing effects of increasingV k: The share of individuals choosingk increases byP k(1−P k) 1

σ; weighting byπk

i captures the additional profit. On the other hand, increasingV k implies reducingRk (and therebyπk

i ) by the same amount; this applies to the share ofindividuals choosingk, P k, capturing the loss in profit. For these two effects to cancelout,we have to haveπk

i = σ1−P k .

It can be shown that the only equilibrium is a symmetric one, where all insurers choose thesame level of utilityV j = V ∀j. Since, in this case,P k = 1

n, in equilibrium profit per

individual isπki =

n

n− 1σ, (9)

and total profit per insurer is

πk =σ

n− 1. (10)

As is to be expected, more competition leads to lower profits:both, profit per individual,πki , and total profit per insurer,πk, increase inσ and decrease inn.

If σ is small, offering a higher utility level yields a large increase in the share of individuals,because individuals are responsive even to small differences in contracts. This raises theincentive to offer a higher utility level, thereby reducingprofits in equilibrium.

If n is large, each insurer’s market share is small. Offering a higher utility level then attractsindividuals from a large ‘external’ market share1 − P k. This again raises the incentive tooffer higher utility levels, lowering profits. We refer to this as the ‘more competition due toa larger external market share’-effect. This effect plays an important role when risk typesare unobservable.

Note that this external market share1 − P k is confined to the interval[0.5, 1[. The effectof the total number of insurers on profits is therefore ratherlimited: Increasing this numberfrom n = 2 to n → ∞ only cuts profit per individualπk

i in half, see condition (9). Incontrast, the effect ofσ on profit per individual is not bounded. In that sense,σ is the moreimportant variable to capture large differences in the level of competition.

2.3 Graphical representation of the equilibrium with one risk type

We will now present the solution graphically in somewhat greater detail than necessary forthis basic model, because it greatly facilitates the derivation of the results for the case ofunobservable risk types.

8

As P k denotes the share of all individuals choosing insurerk, it can be considered a distri-bution functionP k(V k). In equilibrium, when all the other insurers offer the same level ofutility V , we have

P k = P k(V k|σ, V ) =e

V k

σ

eV k

σ + (n− 1)eVσ

. (11)

The shape of this distribution function and of the corresponding densityP k(1 − P k) 1σ

isshown in Figure 2.

0

0.25

0.5

0.75

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

σ = 0.02σ = 0.05σ = 0.10σ = 0.15

(a) Distribution function

0

5

10

15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

σ = 0.02σ = 0.05σ = 0.10σ = 0.15

(b) Density function

Figure 2: Distribution functionP k(V k|σ, V ) and density functionP k(1−P k) 1σ

with n = 2

andV = 1 for different values ofσ

We can depict this distribution functionP k in them-R-diagram that shows the equilibriumwhere all insurers offer{m∗, V } by drawing a shaded area around theI V -indifference curverepresenting the corresponding densityP k(1 − P k) 1

σ, see Figure 3; the different levels of

darkness of this shaded area are a measure of the level of thisdensity.21

If one insurerk charges a higher premium and thereby offers a utility levelV k < V , thecorresponding indifference curveIV

klies aboveI V , see Figure 3 again. As contractA

is above the shaded area, bothP k and the corresponding density are zero.22 Increasingutility V k then moves contractA (along the linem = m∗) into the shaded area, whichincreasesP k and decreasesπk

i . These two effects cancel out when contractA lies on the

I V -indifference curve. IncreasingV k even further then increasesP k beyond1n

, and as soonas contractA is below the shaded area,P k = 1.

For the following reason, this shaded area could be referredto as an ‘indifference curvearea’: Consider the case thatn = 2, so that there is only one other insurerj that offersV .Insurerk, to be chosen by individuali, has to offer a utility level

V k > V j + (εij − εik).

21As a technical detail, note that forn = 2, the maximum of this density is atV k = V , but forn > 2, it is

atV k > V . Therefore the ‘center’ of the shaded area is at theI V -indifference curve forn = 2, and somewhatbelow it forn > 2. To simplify the exposition in the graphs, we will always draw the center of the shaded areaat V .

22In Figure 2, contractA could be, e.g., atV k = 0.1. Of course, strictly speaking,P k > 0 ∀ V k, see (11),but above the shaded area, bothP k and the densityP k(1−P k) 1

σare extremely small and almost equal to zero.

9

A

IVk

I V

pm∗ + nn−1σ

nn−1σ

m∗

R

m

Figure 3: Equilibrium contract{m∗, pm∗ + nn−1σ} in the discrete choice model with

one risk type. The shaded area around the indifference curveI V represents the densityP k(1− P k) 1

σof the distribution functionP k.

For some individuals,εij − εik > 0, so the indifference curves insurerk must offer to make

these individuals indifferent between the two insurers arebelow theI V -indifference curve;if εij − εik < 0, it suffices to offer an indifference curve aboveI V . From the perspective ofinsurerk the shaded area therefore also represents the whole set of all indifference curves,i.e., an ‘indifference curve area’.23

There are two effects ifσ increases: First, the iso-profit line associated with the equilibriumcontract is shifted upwards. Secondly, it is straightforward to show that the distributionfunctionP k as stated in (11) increases forV k < V and decreases forV k > V ; it becomesless steep atV k = V , so the density decreases aroundV (see Figure 2). Ifσ increases, thedistribution function is spread out (over a wider range), which can be depicted in Figure 3by drawing a wider (and lighter) shaded area around the indifference curveI V .

Finally note that if insurerk moves its contract along theIVk-indifference curve,P k does

not change, regardless of whetherIVk

is above, within or below the shaded area. This isbecause the distance betweenIV

kandI V in theR-direction is the same for all levels ofm.

TheIVk-indifference curve is therefore also an iso-P k-curve.

2.4 The equilibrium with two observable risk types

We now turn to the case that there are two risk types,r = H,L, with pH > pL; the share ofL-types isλ. We denote the insurers offering a contract for theL-types as insurers of typeA, and insurers offering a contract for theH-types as insurers of typeB.24 The number of

23If there is more than one other insurer, the argument is the same if εij is replaced bymaxj 6=kεij .24The following Section 3.1 will make clear why do not index insurers byL andH .

10

insurers isnA andnB respectively, withnA + nB = n.

It follows immediately from what we derived for the case of one observable risk type that, inequilibrium, all insurers will offer the efficient level of medical servicesmA = mB = m∗,but premiums will differ according to risk type. Because insurers can decide whether to beof typeA or typeB, total profits per insurer have to be the same for both types ofinsurers,i.e.πA = πB. These profits are given by

πA = λσ

nA − 1and πB = (1− λ)

σ

nB − 1, (12)

and are equal for

nA = λn+ (1− 2λ) and nB = (1− λ)n− (1− 2λ). (13)

Of course,nA andnB have to be integer numbers, so that the expressions given in (13) areonly an approximation to the true value.25 As it is not important for the derivation of ourmain results, we do not elaborate on whethernA as given by (13) has to be rounded up oroff.

However, the requirement ofnA andnB to be integer can, for some parameter settings,cause an equilibrium not to exist: For some values ofnA andnB, it may be profitable for aninsurer of typeB to enter the market for theL-types and become an insurer of typeA; butafter the new ‘equilibrium’ has been attained, whereπA

i is decreased andπBi increased, the

same insurer may then find it profitable to become of typeB again. Following Newhouse(1996), assuming a small fixed cost of setting up a new contract (in this case for switchingfrom one insurer type to the other) then stabilizes the equilibrium.26

As is apparent from condition (13),nA andnB do not depend onσ, the level of competition.This, however, is different for the case of unobservable risk types.

3 Two unobservable risk types

Under perfect competition, for the separating equilibriumto exist, the share ofL-types mustbe below a critical level (Rothschild and Stiglitz 1976). The same applies for this discretechoice model if the level of competition is high (or intermediate). In this case, the argumentfor the non-existence of an equilibrium is the same as under perfect competition: If theshare ofL-types is too large, the ‘separating equilibrium’ can be destroyed by offering acontract that would be chosen by both risk types and yield a higher profit than either ofthe two contracts in the ‘separating equilibrium’. Such a ‘pooling equilibrium’ can then bedestroyed by offering a contract chosen only by theL-types.

25Note that according to (13), the share of insurers of typeA equals the share ofL-types only ifλ = 1

2. For

λ < 1

2, we havenA > λn. This is because withλ < 1

2, there will be fewer insurers of typeA than of typeB,

(nA < nB), so the market served by insurers of typeA will be less competitive. This, c.p., causes profits perindividual to be higher in the smaller market, which inducesa somewhat higher number of insurers to becomeof typeA than given byλn.

26In the large number of simulations where we determined the equilibrium values explicitly, the problem ofnon-existence of an equilibrium only occurred for a small fraction of parameter combinations.

11

This is different if the level of competition is low; in this case, the pooling equilibrium isstable and emerges irrespective of the level ofλ.27 Table 1 summarizes which equilibriaoccur.

Table 1: Type of equilibrium for different levels of competition and different shares of lowrisk types.

perfect competition high/intermediate level of comp. low level of competition

(σ = 0) (σ small/intermediate) (σ large)

λ low enough separating equilibrium separating equilibrium pooling equilibrium

λ too high no equilibrium no equilibrium pooling equilibrium

In the following Section 3.1, where we want to analyze the separating equilibrium, wetherefore assume bothλ andσ to be low enough so that this equilibrium exists. We beginwith the case of a very low level ofσ, so that overall profits are small and the contractdesignated for theL-types yields a negative profit when chosen by anH-type. The effectsof an increase inσ are then derived in Section 3.2 and the pooling equilibrium is discussedin Section 3.3.

3.1 The separating equilibrium for a low level ofσ

If the risk type is unobservable, a contract offered by insurer A (or B) may be chosen byboth risk types.28 Therefore, for both types of insurers, risk type specific utility levels,probabilities and profits have to be defined.

For insurerA, the utility level associated with a contract{mA, RA} depends on the risktyper = L,H according to

V Ar = prv(mA)−RA.

The probability that an individual of risk typer chooses insurerA is given by

PAr =

eV Arσ

eV Arσ +

∑j 6=A e

Vjrσ

. (14)

Finally, type specific profits in terms ofV AL andmA are

πAr = pLv(mA)− V A

L − prmA. (15)

V Br , PB

r andπBr are defined equivalently.

We formulate the objective of insurerA in terms ofV AL andmA, and expressV A

H as

V AH = V A

L + (pH − pL)v(mA). (16)

27We explain why this is the case in Section 3.3.3.28In the following we will often use the term ‘insurerA’ instead of ‘one of the insurers of typeA.’

12

Using these definitions, insurerA’s objective can be stated as

maxV AL,mA

πA = λPAL πA

L + (1− λ)PAHπA

H , (17)

with FOCs

∂πA

∂V AL

= λ

[PAL (1− PA

L )

σπAL − PA

L

]+ (1− λ)

[PAH (1− PA

H )

σπAH − PA

H

]= 0 (18)

∂πA

∂mA= λPA

L

[pLv′(mA)− pL

]+ (1− λ)PA

H

[pLv′(mA)− pH

](19)

+(1− λ)PAH (1− PA

H )

σ(pH − pL)v′(mA)πA

H = 0,

and likewise for insurerB.29 In addition, we have to haveπA = πB, i.e.

λPAL πA

L + (1− λ)PAHπA

H = λPBL πB

L + (1− λ)PBH πB

H . (20)

In equilibrium, when all insurers of typeA offer the same contract for theL-types, and allinsurers of typeB offer the same contract for theH-types, we have

PAL =

eV ALσ

nAeV ALσ + nBe

V BLσ

. (21)

The other market shares,PAH , PB

L andPBH , are defined accordingly.

We will first present the separating equilibrium graphically and then show how the solutioncan be derived from the four FOCs and the profit equality constraint.

With unobservable risk types and perfect competition, in Figure 4, the equilibrium consistsof contractB, chosen by theH-types, and contractA1, chosen by theL-types.30 However,as the shaded area around theIV

BH -indifference curve shows, under imperfect competition,

insurerA would find a considerable share ofH-types choosing contractA1.31 Therefore,contractA1 has to be shifted outside the shaded area.

Assume, that it is shifted (along the iso-πAL -line) toA2, where (almost) none of theH-types

choose this contract. But then insurerA could move its contract along theIVAL -indifference

curve to the right: This would leave the number ofL-types choosing this insurer unaffected(see the definition ofPA

L in (14)), but increase profits perL-type,πAL , because the slope of

theIVAL -indifference curve is larger than the slope of the iso-πA

L -lines for all contracts withmA < m∗. It would also increase the number of theH-types choosing insurerA; however,since the densityPA

H (1 − PAH ) 1

σis (almost) zero at contractA2, at the boundary of the

shaded area this effect is of second order. There is a third effect when moving alongIVAL :

Depending on whether the slope of theIVAL -indifference curve is smaller or larger than the

29As insurerB offers a contract for theH-types, we formulate its objective in terms ofV BH , (notV B

L ).30In this case, the iso-profit lines would of course start at theorigin, asσ = 0.31Here, the shaded area represents the density of the distribution functionPA

H (V AH ) = PA

H (V AL ,mA), or the

‘indifference curve area’ of theH-types from the perspective of an insurer of typeA.

13

m

nB

nB−1σ

R

mA

A3

B

A1

A2

IVBH

IVAL

m∗

Figure 4: Separating equilibrium with two unobservable risk types. ContractsB andA3 areoffered. The casenA = nB, i.e.λ = 0.5, is depicted; fornA 6= nB, the iso-profit lines startat different points on the ordinate.

slope of the iso-profit lines for theH-types,pH , this will increase or decrease profit perH-type,πA

H .

InsurerA will therefore move its contract along theIVAL -indifference curve until these three

effects – the increase ofπAL , the increase ofPA

H , and the change ofπAH – cancel out, which

will be at a contract as indicated byA3.

In equilibrium, a small share ofH-types chooses contractA. This contrasts with the contractoffered by insurerB: As contractB is far away from the shaded area that can be drawnaround theIV

AL -indifference curve, none of theL-types choose contractB.32 As there is

no interference of theL-types, contractB is at the efficient level, as in the case of perfectcompetition.

Result 1. In the separating equilibrium, ifσ is small, then only the benefit package for theL-types is distorted:mA < m∗ andmB = m∗. A small share of theH-types chooses thecontract designated for theL-types, but none of theL-types choose the contract designatedfor theH-types:PA

H > 0 andPBL = 0.

In the remainder of this section we show how these results arereflected in the FOCs. Webegin with insurerB: With PB

L = 0, the FOC with respect tomB simplifies tov′(mB) = 1,

somB = m∗. For the FOC with respect toV BH , we haveπB

H =PBH

1−PBH

σ. For insurerB,

both FOCs are identical to the case that risk types are observable.

This is different for insurerA: If PAH was equal to zero, condition (19) would simplify

to v′(mA) = 1, so we would havemA = m∗. This, together with the lower premium,would induce at least some of theH-types to choose insurerA, a contradiction toPA

H =

32The shaded area around theIVAL -indifference curve represents the density ofPB

L (V BL ) = PB

L (V BH ,mB).

14

0, so PAH > 0. With PA

H > 0, condition (19) is violated forv′(mA) = 1, becauseboth [pLv′(mA) − pL] andπA

H (in the second and third summand) are negative; therefore,v′(mA) > 1, somA < m∗.

The FOC with respect toV AL , condition (18), would simplify toπA

L =PAL

1−PAL

σ for PAH = 0.

However, withPAH > 0, the second bracket is negative becauseπA

H < 0; therefore,πAL

has to be larger than for the case ofPAH = 0, so the equilibrium contractA3 is above the

iso-profit line as shown in Figure 4.

As in this section we consider the case thatσ is very small,PAH will be close to zero, because

the densityPAH (1 − PA

H ) 1σ

will already be large for a small value ofPAH .33 Therefore,A3

will only be slightly above the iso-profit line as shown in Figure 4. AsπAL is almost not

affected by the very low share of high risks, the number of insurers of typeA and typeB,nA andnB, will then not be different from the case when risk types are observable. This,however, changes asσ increases.

3.2 The dependence of the separating equilibrium on the level of competition

So far, the equilibrium under imperfect competition looks rather similar to the case of per-fect competition. We will now show that this only holds for high levels of competition. Inthe following Section 3.2.1, we analyze the effects of a decrease in competition due to anincrease inσ; we discuss a decrease of competition due to a decrease ofn in Section 3.2.2.

3.2.1 The dependence of the separating equilibrium onσ

In Section 2.2 it was shown that an increase inσ increases profits, as insurers reduce theutility levels they offer by increasing the premium, which shifts the iso-profit line associatedwith the equilibrium upwards. The same applies in this separating equilibrium. However,because of the following additional effects, the increase in premiums alone does not yetconstitute the new equilibrium:

Effect onmA

If σ increases, the shaded area around theIVBH -indifference curve becomes wider; contract

A3, if not moved, would be closer to the center of this area (relative to its boundaries), soPAH would increase.34 To avoid being chosen by these additionalH-types, which incur a

negative profit, insurerA has to reducemA. On the other hand, there is the countervailingeffect that as premiums increase, insuring an additionalH-type now causes a smaller loss,which creates an incentive to increasemA.

For a general utility functionv(·), the aggregate of these two effects onmA for a particularlevel of σ is indeterminate. However, ifσ is very small,PA

H is close to zero, whileπAH is

33This also follows immediately from condition (19), which, if σ is close to zero, can only be satisfied forPAH close to zero.

34This also follows directly form the definition ofPAH , which increases inσ for all valuesV A

H < V BH .

15

far below zero. The relative increase ofPAH then outweighs the relative increase ofπA

H , andmA decreases.35 Graphically, in Figure 4, ifσ = 0, the shaded area corresponds to theIV

BH -indifference curve; ifσ becomes larger than zero, the shaded area becomes wider than

the indifference curve, and contractA has to be shifted to the left, irrespective of any effectonπA

H .36

If σ increases,πAH increases and gets closer to zero. As the loss incurred by theH-types

approaches zero (and eventually even becomes a profit), the incentive to avoid being chosenby theH-types is greatly reduced (and eventually vanishes), so that at some point, insurerA increasesmA.

Effect onPAH

The share ofH-types choosing insurerA increases inσ. The proof is given in Appendix A.1;here, we only provide a brief intuitive explanation: If, (because of the wider shaded area),mA was reduced to a level so thatPA

H was the same as before the increase inσ, there wouldthen be an incentive to increasemA, and therebyPA

H , for three reasons: First, because ofthe lower value ofmA, v′(mA) is increased; withPA

H at the same level as before, condition(18) is not satisfied anymore andmA has to be increased. Secondly, ifPA

H is at the samelevel as before, the densityPA

H (1 − PAH ) 1

σis now lower (due to the larger value ofσ), so

that moving along theIVAL -indifference curve does not attract as manyH-types as before.

Thirdly, πAH is increased, so attracting an additionalH-type now causes a smaller loss.

Effect onnA

If σ increases, profits increase faster for type-A insurers than for type-B insurers, so thatat some point it will be profitable for one of the type-B insurers to switch and to become atype-A insurer. The proof can be found in Appendix A.2; here again, we only provide anintuitive explanation: We just showed thatPA

H increases inσ, so the number of individualschoosing any of the type-B insurers decreases. This is the first effect reducing total profitsof type-B insurers relative to type-A insurers.

In addition, as the number of individuals choosing the type-A insurers increases, for thetype-B insurers there is the ‘more competition due to a larger external market share’-effect,which, as we saw in Section 2.2, decreases profits per individual. Due to these two effects,πB increases at a lower rate thanπA, so that at some level ofσ, the first of the type-B-insurers finds it profitable to become an insurer of typeA, andnA increases. Ifσ increasesfurther, the second type-B insurer switches, and so on, until at some level ofσ the last ofthe type-B insurers becomes a type-A insurer, and the pooling equilibrium emerges.37

35It is straightforward to show that withmA held fixed, bothPAH and the densityPA

H (1− PAH ) 1

σincrease in

σ for σ close to zero;v′(·) then has to be increased, so that condition (19) is still satisfied.36Note that the decrease ofmA is not necessarily confined to a small interval ofσ close to zero, but can occur

for a wide range ofσ, see Section 3.5.37Note that a single insurer of typeB can not charge an excessively high premium, because this insurer would

lose its insured to the insurers of typeA.

16

Effect onmB

We finally discuss why for intermediate levels of competition, contractB is distorted. Ifσincreases, the shaded areas around both indifference curves get wider. At some level ofσ,the shaded area around the indifference curve of theL-types,IV

AL , becomes so wide that it

‘reaches’ contractB, so that a small share of theL-types chooses contractB (see Figure 5,where only the shaded area around theIV

AL -indifference curve is drawn). It will then be

profitable for insurerB to move its contract along theIVBH -indifference curve and reduce

mB . This leaves the share of theH-types choosing this insurer unaffected, but increases theshare of theL-types, (as the iso-PB

L -curves have a lower slope than theIVBH -indifference

curve), thereby increasing profits. Of course, this also reduces profits perH-type,πBH , but at

(or close to) the efficient level ofm, this effect is of second order.38 The larger the densityPBL (1 − PB

L ) 1σ

around contractB, the larger the distortion of this contract. Therefore,

as long as the widening of the shaded area around theIVAL -indifference curve leads to an

increase of the density at contractB, the distortion will increase inσ.

m

R

mA

B0

B1

IVBH

IVAL

m∗

Figure 5: Separating equilibrium with two unobservable risk types andσ large: ContractBdistorted fromB0 toB1.

Comparing the effects of a decrease in competition onmA andmB , there is an asymmetryin that an increase inσ changesmA even for low values ofσ, while the effect onmB onlyarises above some threshold level ofσ, at which the shaded area aroundIV

AL ‘reaches’

contractB.

Result 2. In the separating equilibrium,mA first decreases and then increases inσ. Forintermediate levels ofσ, both benefit packages are distorted:mA < mB < m∗; in addition,a share of both risk types chooses the contract designated for the other risk type:PA

H > 0and PB

L > 0. The number of insurers offering the contract designated for the L-typesincreases inσ.

38Of course, strictly speaking,mB is always distorted, asPBL is always larger than zero. However, for low

levels ofσ, PBL is so close to zero, that the distortion ofmB is negligible. In the example we present in

Section 3.5,PBL is on the order of10−30 for low levels ofσ.

17

3.2.2 The dependence of the separating equilibrium on the total number of insurers

If the total number of insurersn increases,nA andnB increase proportionally forλ = 12 ,

and almost proportionally forλ 6= 12 . Accordingly, all market shares decrease (about) pro-

portionally, which leaves condition (19) unchanged. Also,there is no widening of theshaded areas around the indifference curves;39 there is only an effect on on profits: Ifnincreases, profits per individual go down. This increases the loss caused by anH-type, sothe incentive to avoid theH-types increases. This is reflected in condition (18) wheremA

has to be decreased whenπAH decreases. Therefore,mA decreases inn.

Result 3. In the separating equilibrium, the distortion of the benefitpackage of the low risktype increases in the total number of insurers:∂mA

∂n< 0.

3.3 The pooling equilibrium

3.3.1 The dependence of the pooling equilibrium on the levelof competition

As has been shown in Section 3.2.1, ifσ increases, the number of insurers of typeA goesup. At some point, the last insurer of typeB becomes an insurer of typeA, and a poolingequilibrium occurs.40 Using the fact that in this casenB = 0 andPA

L = PAH = 1

nA , wherenA = n, condition (18), the FOC with respect toV A

L , simplifies to

λπAL + (1− λ)πA

H =nσ

n− 1. (22)

Solving forπAH and substituting in (19), the FOC with respect tomA, we have

[1−

λ(1− λ)(pH − pL)2

nσn−1p

mA

]v′(mA) = 1. (23)

Because the fraction in (23) is positive, it is immediately apparent thatv′(mA) > 1, so thatmA is distorted downward. As is to be expected, the distortion increases in the differencepH−pL. Also, it decreases inσ and increases inn: The distortion in the pooling equilibriumis less severe if the market is less competitive.

Result 4. In the pooling equilibrium, the distortion increases in thelevel of competition:∂mA

∂n< 0 and ∂mA

∂σ> 0.

3.3.2 Comparison of the pooling equilibrium with the Wilson-equilibrium

In general, this pooling equilibrium does not coincide withthe Wilson-equilibrium (Wilson1977), which consists of the contract on the pooling zero-profit line that maximizes the util-

39The distribution functionP k depends on the utility levels offered by the other insurers only via the aggre-

gate∑

j 6=ke

Vk

σ . If this aggregate increases (e.g., due to an increase inn), this shifts the distribution functionto the right (in Figure 2), but does not change its shape.

40To keep the notation simple, we do not introduce an additional index for the pooling equilibrium but denoteall insurers to be of typeA.

18

ity of theL-types.41 We will denote this contract, as it would not be the Wilson-equilibriumif the share ofL-types was low enough for the separating Rothschild-Stiglitz-equilibriumto exist, simply as the Wilson-contract.

Formally, for the Wilson-contract,m satisfiespLv′(mWI) = p. Using conditions (22) and(23), it is straightforward to show that for the discrete choice model, this requiresπA

H = 0,see Appendix A.3. Of course, if profits for theH-types are zero,H-types do not play arole when choosing the optimal contract on the iso-profit line, so insurers will maximize theutility of the L-types (to have as manyL-types as possible).

However, ifπAH < 0, it will be profitable to reducemA belowmWI : this will only have

a second order effect on the utility of theL-types, but a first order effect of reducing thenumber ofH-types. In this case,mA < mWI . If, on the other hand,πA

H > 0, then havingmoreH-types increases profits, so insurers will raisem abovemWI .

Result 5. The pooling equilibrium only coincides with the Wilson-contract if profit perH-type is zero:mA T mWI for πA

H T 0.

3.3.3 Stability of the pooling equilibrium

From a technical perspective, this result shows that in a Rothschild-Stiglitz model, a poolingequilibrium as a Nash-equilibrium in pure strategies can exist if there is imperfect compe-tition. The pooling equilibrium can therefore be rationalized without imposing Wilson-foresight, a concept that has been criticized by Rothschildand Stiglitz (1997).

Newhouse (1996) had already identified a reason for a poolingequilibrium to exist, fixedcosts of setting up a new contract: If trying to attract theL-types with a new contract causeshigh costs, the pooling equilibrium is stable.

Here, the pooling equilibrium is stable for a different reason: Offering a contract betweenthe indifference curves of the two risk types would, under perfect competition, only attractall theL-types and thereby destroy the pooling equilibrium. Here, if σ is large, a contractclose to the pooling equilibrium attracts bothL- andH-types, where, due to the large in-fluence of the utility componentεij that is independent of the benefit-premium-bundle, therelative share of theL-types in this new contract is not much larger than in the poolingequilibrium. To only attract theL-types, the new contract would have to be far away fromthe pooling equilibrium, so that it is on the same indifference curve of theL-types, butoutside (i.e. above) the shaded area of theH-types. Such a contract would be associatedwith a much lower premium, and thereby not provide a higher profit than the contract of thepooling equilibrium.

41It is common to refer to the Wilson-equilibrium whenever – because the share of theL-types is too large –the separating Rothschild-Stiglitz-equilibrium does notexist; see Zweifel et al. (2009, p. 178). Of course, weonly compare the level ofm, and not premiums.

19

3.4 Welfare effects of different levels of competition

In this model, welfareW is given as the sum of expected surplus generated by the consump-tion ofm,

W =∑

i

Si =∑

i

(piv(mi)− pimi

), (24)

with mi being the level of medical services consumed by individuali. Of course, the pre-miumR does not appear in (24), as it is only a transfer from the insured to the insurer.

From what has been derived in the previous sections it follows that the welfare effects of adecrease in competition for the separating equilibrium areambiguous, while for the poolingequilibrium, welfare increases.

For the separating equilibrium, welfare decreases inσ if σ is close to zero, becausemA de-creases andPA

H increases, so that a larger number of individuals chooses the benefit packagethat is more distorted. For intermediate levels ofσ, the welfare effects of an increase inσare indeterminate: On the one hand,mA increases, but on the other handmB decreases andPAH andnA increase. The welfare effects of a decrease in competition due to a decrease in

the total number of insurers are indeterminate as well. It was shown that asn decreases,mA increases, because the loss associated with theH-types decreases as competition goesdown. But for the same reasonPA

H increases, creating a countervailing effect on welfare.

For the pooling equilibrium, on the other hand, as competition goes down, welfare unam-biguously increases, see condition (23); this holds for both the increase inσ and the decreasein n.

Result 6. For very high levels of competition, welfare decreases inσ: ∂W∂σ

< 0 for σ closeto zero. For low levels of competition, (so that a pooling equilibrium emerges), welfaredecreases in the level of competition:∂W

∂σ> 0 and ∂W

∂n< 0 for nB = 0.

3.5 Example

We finally illustrate the results with an example, for which we assumen = 10, pL = 0.2,pH = 1, λ = 0.5 andv(m) = ln(m), so that the efficient level of medical services ism∗ = 1 and one of the risk types is chronically ill. The equilibriumvalues for differentlevels ofσ are shown in Table 2; the Rothschild-Stiglitz-equilibriumcan be found in thefirst, the Wilson-contract in the last row.

Under imperfect competition,mA first decreases and then increases inσ; in addition,nA

increases, until the pooling equilibrium is reached atσ = 0.19.42

Even for the lowest level ofσ, some of theH-types choose contractA; the share ofH-typesamong all insured choosing an insurer of typeA then steadily increases inσ. On the otherhand, none of theL-types choose an insurer of typeB until the shaded area around theIV

AL -

indifference curve reaches contractB; this occurs atσ = 0.08, where also the distortion of

42In this example, in all pooling equilibria,mA is above the level of the Wilson-contract. For higher levelsof pL, e.g.,pL = 0.4, in some of the pooling equilibriamA < mWI .

20

Table 2: Example I withv(m) = ln(m), pL = 0.2, pH = 1, λ = 0.5, n = 10, for differentvalues ofσ. The first row (RS) contains the Rotschild-Stiglitz-equilibrium, the last row(WI) the ‘Wilson’-contract.

σ nA nB mA mB A’s share B’s share W

of H-types ofL-types

RS - - .398 1.00 0.0% 0.0% -.632

.01 5 5 .377 1.00 0.5% 0.0% -.636

.02 5 5 .364 1.00 1.2% 0.0% -.640

.04 5 5 .346 1.00 2.8% 0.0% -.647

.06 5 5 .337 1.00 5.1% 0.0% -.654

.08 5 5 .334 .998 7.9% 0.1% -.662

.10 6 4 .324 .994 12.5% 0.1% -.677

.15 7 3 .362 .940 29.7% 1.4% -.717

.17 8 2 .395 .884 39.2% 3.4% -.736

.18 9 1 .418 .847 45.4% 5.0% -.749

.19 pooling .442 - 50.0% - -.755

.25 pooling .510 - 50.0% - -.710

WI pooling .333 - 50.0% - -.859

mB sets in. Ifσ increases further,mB decreases and is way below the efficient level forσ = 0.18.

Regarding welfare, we find that it decreases inσ in the separating equilibrium for all levelsof σ (and not just ifσ is close to zero)), and, of course, increases inσ for the poolingequilibrium.

4 Implications for risk adjustment

We now discuss the implications of the results derived so farfor risk adjustment. In partic-ular, we show that the welfare effects of introducing or improving a risk adjustment scheme(RAS) critically depend on the level of competition: For lowand high levels of compe-tition, a RAS that becomes more precise unambiguously increases welfare; however, forintermediate levels of competition, welfare may initiallyremain constant or even decrease.

We will not model explicitly which risk adjusters are used inthe RAS, or which economet-ric method is applied to estimate the payments. What is important for our model is thatwhenever a RAS becomes more precise, it reduces the cost difference between the two risktypes to a larger extent. A RAS can be improved by, e.g., usingmore and more risk ad-justers, like hospital stays, or diagnostic information, which are more informative signalsfor the risk type than just demographic information. A regulator may also apply the formulafor optimal risk adjustment developed by Glazer and McGuire(2000). In all cases, the cost

21

difference between risk types will be reduced, and with a perfect RAS, this difference iseliminated completely.

We will model the RAS in the easiest way possible: Each insurer receives a payment ofRAH for anH-type, and has to payRAL for anL-type. For the RAS to break even, wehave to have

λRAL = (1− λ)RAH .

SettingRAH to some levelRA, this requiresRAL = 1−λλ

RA. In this way, the RAS can beexpressed with only one parameter,RA. AsRA increases, the RAS becomes more precise.

We will first present an example to show how the welfare effects of increasingRA dependon the level of competition. We then explain why for intermediate levels of competitionwelfare may decrease inRA in the separating equilibrium (Section 4.2), but not in thepooling equilibrium (Section 4.3).

4.1 Example

We present the same example as in Section 3.5, withv(m) = ln(m), pL = 0.2, pH = 1andλ = 0.5, and show the impact on welfare by increasingRA = RAH = RAL from 0to 0.4, at which level the cost difference between theL-type and theH-type is eliminatedcompletely. Results are shown for 10 and 20 insurers (see Figure 6(a) and (b) respectively),for different levels of competition:σ = 0.01 (very competitive),σ = 0.10, σ = 0.12 andσ = 0.14 (intermediate levels of competition), and for the lowest level of σ for which thepooling equilibrium emerges:σ = 0.19 for 10 insurers, andσ = 0.20 for 20 insurers.

The equilibrium values for the level of medical servicesmA andmB and the number ofinsurersnA andnB for one of the cases (n = 20 andσ = 0.12) can be found in Table 4 inAppendix A.4. Here, we only plot the equilibrium levels of welfare as a function ofRA forthese five different values ofσ. The highest level of welfare for this example is 0.6, whichoccurs when all individuals receivem∗ = 1.

Figure 6: Example III withpL = 0.2, pH = 1, λ = 0.5 and different levels ofσ. WelfareW is depicted as a function ofRA, with RA increasing from 0 to 0.40.

-0.76

-0.74

-0.72

-0.70

-0.68

-0.66

-0.64

-0.62

-0.60

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

σ = 0.01

σ = 0.10

σ = 0.12

σ = 0.14

σ = 0.19

(a) 10 insurers

-0.76

-0.74

-0.72

-0.70

-0.68

-0.66

-0.64

-0.62

-0.60

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

σ = 0.01

σ = 0.10

σ = 0.12

σ = 0.14

σ = 0.20

(b) 20 insurers

As can be seen, forσ = 0.01 andσ = 0.19, (or σ = 0.20 in case ofn = 20 insurers),

22

welfare increases monotonously inRA.43 However, for intermediate levels of competition,welfare stays about constant or even decreases as long asRA is below the threshold level,at which the pooling equilibrium is reached; only above thislevel, does welfare increasesin RA.44 For the case of 20 insurers andσ = 0.10, this threshold level is as high asRA = 0.17: Although the RAS reduces the cost difference between the two risk types bymore than 40%, there is no increase in welfare.

For intermediate levels of competition welfare initially does not increase inRA because theRAS not only reduces a distortion (by increasingmA), but also introduces or exacerbatestwo other distortions: As we show in the following section, the share ofH-types choosingthe benefit package designated for theL-types increases inRA; in addition, the distortionof the benefit package for theH-types becomes more severe (mB decreases).

This contrasts with the case of either a low or a high level of competition, where theseadditional distortions do not occur (or are so small that they are negligible); for these levelsof competition, welfare increases inRA.

4.2 Risk adjustment in the separating equilibrium

Taking into account the payments of the RAS, type specific profits for insurerA are

πAL = pLv(mA)− V A

L −1− λ

λRA− pLmA (25)

πAH = pLv(mA)− V A

L +RA− pHmA. (26)

The FOCs for insurerA’s objective are therefore identical to (18) and (19), but with πAL and

πAH now defined by (25) and (26). The same applies to insurerB.

For insurerB, from the FOC with respect tomB it follows that for low values ofσ (sothatPB

L = 0), we havev′(mB) = 1, as before. From the FOC with respect toV BH , we

haveπBH = nB

nB−1σ, again as before. IfRA is increased, so that insurerB receives a larger

subsidy for eachH-type, premiums are reduced (and utilityV BH increased) by the same

amount, so thatπBH stays constant. For insurerB, we can therefore depict an increase in

RA by a decrease inRB of equal size: In Figure 7, the contract offered is shifted fromB0

to B1; accordingly, there is a downward shift of the corresponding iso-profit line and theindifference curve.45

There is an opposite effect on the premium of insurers of typeA: asRA increases, this, c.p.,increases the premiumRA (and reducesV A

L ) byRAL = 1−λλ

RA, shifting the iso-profit line

43Forσ = 0.01, there is a small decrease in welfare for some high level ofRA; this is because at this levelof RA there is a switch from the separating to the pooling equilibrium.

44Note that for these intermediate levels of competition, there is usually one level ofRA for which anequilibrium does not exist: As we already mentioned in Section 2.4, for one of the candidate equilibria(mA, V A

L ,mB , V BH ), one of the insurers of typeB has an incentive to become an insurer of typeA; in the

candidate equilibrium for these new levels ofnA andnB , an insurer of typeA then has an incentive to becomean insurer of typeB. In Figure 6 we plot the higher of the two levels of welfare of the two candidate equilibriato present the case where the RAS is more successful in improving welfare.

45In Figure 7, the case of a very low level ofσ is depicted, so that the two shaded areas do not overlap andcan be distinguished.

23

upwards. Similar to the case of an increase ofσ in Section 3.2.1, this does not yet constitutethe new equilibrium; there will also be an effect onmA.

m

R

∆RA

∆RA

∆RA

mA1mA

0 m∗

A0

A1

B0

B1

IVBH

I VBH

Figure 7: Equilibrium without and with (imprecise) risk adjustment; the caseRAL = RAH ,i.e.λ = 0.5, is depicted.

As can be seen from Figure 7, due to the downward shift of theIVBH -indifference curve

(to I VBH ), and the upward shift of the iso-profit line of insurerA, offering a contract with

the same level ofmA reduces the share ofH-types choosing contractA. This also followsimmediately from the definition ofPA

H , which decreases asV AH decreases andV B

H increases.

This decrease inPAH creates an incentive to increasemA, which can also be seen from the

FOC with respect tomA: rearranging terms, condition (19) reads as

v′(mA)− 1 +1− λ

λ

PAH

PAL

[v′(mA)−

pH

pL

]+

1− λ

λ

PAH

PAL

1− PAH

σ

pH − pL

pLπAHv′(mA) = 0.

(27)

As PAH is reduced,mA has to be increased, so that this condition is satisfied again. In

addition, due to the payments of the RAS for theH-types,πAH is increased, which – in an

equivalent manner as for the case of an increase inσ in Section 3.2.1 – creates a secondincentive to increasemA. AsmA unambiguously increases, this, c.p., leads to an increasein welfare.

The effect onPAH , however, is ambiguous: Assume thatmA is increased to a level so that

PAH is the same as before. At that point, it is not clear whether there is an incentive to

increasemA, and therebyPAH , even further or not. On the one hand,πA

H is increased, buton the other hand,v′(mA) has already been decreased, so for a general utility function, itis indeterminate whether (27) is positive or negative. However, as the effect ofRA onπA

H

is linear, while the effect onv′ is decreasing, it is likely thatPAH increases inRA, if RA is

large.46

46In the large number of examples for which we derived the equilibrium for a particular utility functionexplicitly, PA

H always increased inRA even from the beginning (RA = 0).

24

The increase inPAH , if it occurs, captures the first effect that reduces welfare: EachH-type

choosing contractA instead of contractB induces a loss of welfare, becausemA < mB. Inaddition, ifPA

H increases, we have the same effects on profits as already described in Sec-tion 3.2.1: Due to the loss of individuals, competition among insurers of typeB increases,which reduces profits per individual; together with the smaller market share, total profit perinsurer of typeB decreases. At some point, one of the type-B insurers will switch andbecome a type-A insurer. This is the second negative effect on welfare: Eachinsurer thatswitches to become an insurer of typeA incurs a welfare loss, as all its insured receivemA

instead ofmB .47

There is a third negative effect on welfare that occurs regardless of whetherPAH increases

or not: We saw that asRA increases, this shifts theIVBH -indifference curve downwards,

and theIVAL -indifference curve upwards, so the distance between thesetwo indifference

curves decreases atm∗. This will, in similar manner as described in Section 3.2.1,lead toa distortion ofmB below the efficient level, as soon as the shaded area around the IV

AL -

indifference curve ‘reaches’ contractB.

Result 7. A RAS that becomes more precise reduces the distortion of thelevel of medicalservices for theL-types in a separating equilibrium:∂m

A

∂RA> 0.

However, a RAS that becomes more precise may also decrease welfare because (i)PBH may

decrease, (ii)nB may decrease, and(iii), at some level ofRA, mB decreases below theefficient level.

Whether these three countervailing effects are significant, or only reduce the effectivenessof the improvement of the RAS, of course depends on the specific utility function.

It also depends on the level ofσ: If σ is small, the shaded area around theIVBH -indifference

curve will be small. In this case, the densityPAH (1 − PA

H ) 1σ

will already be large whenPAH

is still small, so for small values ofσ the first countervailing effect is greatly reduced. AsPAH is small, the difference in profitsπA − πB is small (see Appendix A.2), so that none

of the insurers of typeB switch to become of typeA; then the second countervailing effectdoes not exist. Thirdly, ifσ is small, the shaded area around theIV

AL -indifference curve will

be narrow, so it will not ‘reach’ contractB until RA is close to the level, at which the costdifference between the two risk types is eliminated completely; for small and intermediatelevels ofRA, the third countervailing effect does not exist either. Therefore, ifσ is small(andRA not too large), welfare increases as a RAS becomes more precise.

4.3 Risk adjustment in the pooling equilibrium

For the pooling equilibrium, the FOC with respect tomA simplifies to[1−

(1− λ)(pH − pL)[λ(pH − pL)mA −RA]nσn−1p

]v′(mA) = 1. (28)

47Of course, when this insurer switches and becomes a type-A insurer, a large share of theH-types of thisinsurer will choose another insurer of typeB; but those with a high preference for this particular insurer (highεij ) will stay with this insurer, causing the welfare loss.

25

With RA = 0, i.e. without risk adjustment, we have condition (23) from Section 3.3. AsRA increases, the fraction in (28) decreases, somA increases; withRA = λ(pH − pL)m∗,the distortion is eliminated. Forλ = 1

2 , as soon asRA equals half the difference in expectedcosts between the two risk types, the cost difference vanishes, becauseRA both has to bepaid by the insurer for anL-type, and is paid to the insurer for anH-type.

Result 8. For the pooling equilibrium, an increase inRA unambiguously decreases thedistortion of the benefit package and increases welfare:∂mA

∂RA> 0 and ∂W

∂RA> 0

5 Discussion

In this section, we briefly discuss some of the assumptions ofour model and how they mayaffect the results that have been derived.

5.1 Distributional assumption for εij

The model has been explicitly solved only under the assumption thatεij is i.i.d. extremevalue, but we think that the results also hold for different distributional assumptions. Be-cause the main effects could also be explained graphically,the results should be similar aslong as the shaded areas around the indifference curves represent a unimodal density.

For a large number of examples, we determined the equilibrium under various other dis-tributional assumptions forεij and always found the results to be very similar.48 Table 3presents the equilibrium values of the example of Section 3.5 for three distributional as-sumptions ofεij other than the extreme value: the normal, the triangular andthe uniformdistribution.49 Even with a uniform distribution forεij , the density represented by theshaded area is unimodal; (e.g., this density would be triangular forn = 2).

For low values ofσ (see the upper part of Table 3 withnA = nB = 5), the differences arevery small: For all four distributions,mA decreases inσ, whilemB remains at the efficientlevel. AlsoPA

H , the share ofH-types choosing one of the insurers of typeA, is very similarfor all four distributions.

As σ increases so thatnA increases, two differences emerge: First, the levels ofσ at whichnA increases are not identical for the four distributions, seethe lower part of Table 3, wherealways the smallest value ofσ after an increase innA is presented. E.g., the lowest levelof σ so thatnA = 6 is 0.10 for the extreme value distribution; it is somewhat higher at0.14 and0.15 for the normal and the triangular distribution, and considerably higher for theuniform distribution at 0.21. However, this difference does not seem to be important.

Secondly, the distortion ofmB is much smaller for the other three distributions. This isbecause for a given level ofσ, the shaded area around the indifference curves is widest forthe extreme value distribution; as this distribution has fatter tails, the shaded area around theIV

AL -indifference curve ‘reaches’ contractB for a lower level ofσ than is the case for the48The Gauss code is available from the author upon request.49Note that for all four distributions, the variance is given asVar(εij) = σ2 π2

6.

26

Table 3: Example I withv(m) = ln(m), pL = 0.2, pH = 1, λ = 0.5, n = 10 for differentdistributional assumptions

extreme value normal triangular uniform

nA nB σ mA mB σ mA mB σ mA mB σ mA mB

5 5 .01 .377 1.00 .01 .384 1.00 .01 .386 1.00 .01 .387 1.00

5 5 .02 .364 1.00 .02 .373 1.00 .02 .376 1.00 .02 .377 1.00

5 5 .04 .346 1.00 .04 .358 1.00 .04 .360 1.00 .04 .363 1.00

5 5 .06 .337 1.00 .06 .346 1.00 .06 .349 1.00 .06 .351 1.00

5 5 .08 .334 .998 .08 .340 1.00 .08 .340 1.00 .08 .343 1.00

6 4 .10 .324 .994 .14 .330 1.00 .15 .332 1.00 .21 .351 .996

7 3 .15 .362 .940 .18 .355 .994 .19 .361 .999 .28 .363 .981

8 2 .17 .395 .884 .20 .380 .982 .21 .380 .997 .33 .389 .954

9 1 .18 .418 .847 .22 .403 .973 .23 .406 .993 .38 .406 .940

pooling .19 .442 .442 .23 .421 .421 .24 .416 .416 .42 .423 .423

other distributions. In technical terms, the (excess) kurtosis is largest for the extreme valuedistribution: kev = 2.4; it is considerably smaller for the normal (kn = 0.0), the triangular(ktr = −0.6) and the uniform distribution (ku = −1.2). The higher the kurtosis, the higherthe distortion ofmB (for a given level ofσ).

On the other hand, the levels ofmA are very similar for the four distributions, as is thelevel ofm when the pooling equilibrium is reached. Also, for each of the four distributions,welfare decreases inσ for the separating equilibrium, and increases inσ for the poolingequilibrium.

5.2 Conditional Logit vs. Nested Logit

At first glance, it may appear as if for an individual who chooses an insurer of typeA,another type-A insurer is a closer substitute than a type-B insurer, so that a nested logitmodel may seem more appropriate than the simple conditionallogit that we considered.

From the perspective of an econometrician, this is certainly true, because, a priori, it cannotbe ruled out there there are also some unobserved factors that are more alike among type-A insurers than between type-A and type-B insurers. Therefore, the econometrician willsimply test whether a nested logit model applies.

Here, however, we want to explicitly analyse the effects that arise due to the differencesin the benefit packages. Assuming, in addition, that there are also some unobserved fac-tors which are equal among the type-A insurers, i.e. assuming some non i.i.d.-error termstructure, would only obscure the effects we are interestedin.

Regarding the IIA assumption that is implied by the logit model, the famous red bus-blue

27

bus problem50 does not occur in our setting, because we explicitly model two different risktypes.51 Consider, e.g., the case ofλ = 0.5 and four insurers: Withσ small enough, twoinsurers will be of typeA, each covering half of theL-types, and a small share of theH-types, say 1% (i.e. 0.5% of the entire market); the other two insurers will be of typeB, eachcovering about half of theH-types. Each insurer of typeA will therefore cover 25.5% ofthe entire market, and each insurer of typeB 24.5%. If we now add two more insurers oftypeA, these four type-A insurers will not cover two thirds of the entire market, as inthered bus-blue bus example. Instead, allL-types are evenly distributed among the four type-A

insurers; in addition, the third and fourth type-A insurer will cover about the same share ofH-types as the first and the second type-A insurer (1% of theH-types, or 0.5% of the entiremarket). Therefore, each insurer of typeA will cover about14 ·50%+0.5% = 13%, and theaggregate market share of all type-A insurers will only increase from 51% to about 52%.

5.3 Premium set by regulator

We formulated the model inm-R-space, and not inm1-m2-space withR set by a regulator,as was the setting of Glazer and McGuire (2000). We did this tonot obscure the welfareeffects of different levels of competition. Ifσ increases, profits go up, so a regulator wouldhave to increaseR. However, as we saw in Section 3.2.1, for the case of unobservable risktypes, profits for the two types of insurers increase at different rates. Therefore, it is notclear at which rate the regulator would have to increaseR to not affect welfare.

Nevertheless, all results regarding the distortions of thebenefit packages are easily trans-ferred intom1-m2-space. There, a distortion always consists of a too low level of m1 (ifs = 1 is the illness for which there is heterogeneity in risk) and atoo high level ofm2,see Glazer and McGuire (2000). The shaded areas would then have to be drawn around theindifference curves inm1-m2-space, but the arguments for the different effects would bethe same.

6 Conclusion

We have analyzed the interaction of imperfect competition and adverse selection in healthinsurance markets. Within a discrete choice setting which endogenises whether a separat-ing or a pooling equilibrium emerges, the following main results have been derived: In aseparating equilibrium, for intermediate levels of competition, both benefit packages aredistorted. As the level of competition decreases, the distortion decreases for the low risktype, but increases for the high risk type; in addition, the number of insurers offering thecontract for the high risk type decreases, until a pooling equilibrium is reached. The poolingequilibrium may be below, at, or above the ‘Wilson’-contract.

We also showed that although each individual has the same tendency to ‘make mistakes’ bynot choosing the contract that is most favorable in terms of medical services and premium,

50See Train (2009, p. 46).51Because there is more than one risk type, the model is actually a (rather degenerate) mixed logit, see Train

(2009), Chapter 6.

28

in equilibrium there is an asymmetry in that it is primarily the high risk which choose the‘wrong’ contract.

Finally, we showed that under imperfect competition there is no clear-cut distinction be-tween the separating and the pooling equilibrium: for intermediate levels of competition,each of the two contracts of the separating equilibrium is chosen by both risk types. If thelevel of competition decreases, these ‘two pooling contracts of the separating equilibrium’become more alike, until for a low enough level of competition the pooling equilibrium withonly one contract emerges.

We also determined the implications of imperfect competition on the effectiveness of arisk adjustment scheme. For intermediate levels of competition we identified three welfaredecreasing effects that can occur if an imprecise RAS is onlyimproved to a small degree. Ifthese effects are of economic importance, it is even more important for a regulator to use aRAS that reduces the cost differences between risk types to alarge degree, so that one canbe confident that the RAS creates the positive welfare effects it is implemented for.

The theoretical model we presented complements a number of very recent empirical studieswhich analyze adverse selection in health insurance markets with a focus on inefficientpricing of agivenset of benefit packages. These studies have found that the welfare lossescaused by inefficient pricing are surprisingly low.52 However, as explicitly stated by Einav,Finkelstein, and Levin (2010), the welfare losses due to an inefficient set of benefit packagesmay be much larger than the welfare losses due to inefficient pricing. Our model focuses onthese inefficiencies caused by the distortions of the benefitpackages. We showed that eventhe most generous benefit package offered may not represent the efficient level of medicalservices, and that too few insurers offer this benefit package. If these additional distortionsexist and are of economic importance, the overall welfare losses caused by adverse selectionmay therefore indeed be considerably higher than those onlycaused by inefficient pricing.

52See Einav, Finkelstein, and Cullen (2010), Bundorf et al. (2012) and Handel (2013).

29

A Appendix

A.1 Proof that PAH increases inσ

We will first consider a small, noninfinitesimal increase inσ by ∆σ > 0, which allowsto depict some of the effects graphically; we can then let∆σ become arbitrarily small(∆σ → 0). A ∼ is used to indicate all variables after the increase ofσ, so, e.g.,σ = σ+∆σ.

Denote bySAL andSB

H the surplus generated bymA andmB for the respective risk type, i.e.

SAL = pLv(mA)− pLmA and SB

H = pHv(mB)− pHmB . (29)

Using the FOC with respect toV BH , i.e.πB

H = σ1−PB

H

, andπBH = SB

H − V BH , we have

V BH = SB

H −σ

1− PBH

. (30)

Since for low levels ofσ, mB = m∗, and therefore does not depend onσ, we have

SBH = SB

H , so ∆SBH = 0.

If PBH did not change,∆V B

H would be given by

∆V BH = −

∆σ

1− PBH

. (31)

In Figure 8, this decrease ofV BH is depicted by the movement of insurerB’s contract from

B0 toB1.

For insurerA, rewrite the FOCs with respect toV AL andmA as

λ

[PAL (1− PA

L )

σπAL − PA

L

]+(1−λ)

[PAH (1− PA

H )

σπAH − PA

H

]= 0 (32)

[λpLPA

L + (1− λ)pLPAH + (1− λ)(pH − pL)

PAH (1− PA

H )πAH

σ

]v′(mA) (33)

= λpLPAL +(1−λ)pHPA

H .

For insurerA, condition (33) can be considered as implicitly defining a functionmA(V AL ),

which for each level ofV AL determines the optimal level ofmA. Likewise, condition (32)

implicitly defines a functionV AL (mA). The loci of these two curves of course pass through

A0, the contract offered by insurerA before the increase ofσ.

With contractA0, insurerA will have a certain share ofH-types,PAH . The set of all the

benefit-premium-bundles with which insurerA attracts this share ofH-types constitutesthe iso-PA

H -curve; it has the same shape as theIVBH -indifference curve, shifted upwards; see

Figure 8.

30

m

R

A5

A4

A3

B1

B0

A0

A1

A2

PAH(σ)

I VBH

PAH(σ)

IVBH

mA m∗

Figure 8: Equilibrium for two different values ofσ

If σ increases,V BH is reduced; this shifts theIV

BH -curve upwards, and with it the iso-PA

H -curve. The distance between the two iso-PA

H -curves is larger than the distance between thetwo IV

BH -curves, because asσ increases, the shaded area around the indifference curves

becomes wider. To leavePAH unchanged, insurerA would have to offer a contract on the

new iso-PAH -curve, which is denoted byPA

H (σ) in Figure 8.

It is now argued that the new contract chosen by insurerA will be to the right of this newiso-PA

H -curve, so that in equilibriumPAH increases. To do so, it will be shown that the locus

of the functionmA(V AL ), implicitly defined by (33) withσ increased, is partly to the right

of the new iso-PAH -curve, and that the new contract is exactly on this part ofmA(V A

L ).

Consider first, that insurerA offers contractA1, which is on the same iso-πAH -line as con-

tractA0. With A1, in (33) all variables except formA andσ are at the same level as before.Becauseσ has been increased, which increases the bracket, and because mA has been re-duced, which increasesv′(mA), the left hand side of condition (33) is now larger than theright hand side; therefore,mA has to be increased, which increasesPA

H .

Consider now, instead, contractA2, which has been chosen so thatπAH

σ=

πAH

σ. At A2,

the bracket on the LHS of (33) attains the same value as beforethe increase ofσ. At allpoints on the new iso-PA

H -curve aboveA2, the bracket is larger than before. In addition, formA < mA, we havev′(mA) > v′(mA). Therefore, for all points on the new iso-PA

H -curvebetweenA2 andA3, the LHS of (33) is larger than the RHS, somA has to be increased,which increasesPA

H . Condition (33) could only be satisfied for a point belowA2, or above

31

A3. If such a point belowA2 or aboveA3 did not exist, the locus of the functionmA(V AL )

would always be to the right of the new iso-PAH -curve; in this case, it follows immediately,

thatPAH is increased. We therefore consider the case that these points do exist.

Assume first, condition (33) is satisfied for a point belowA2. At such a point, we would

haveπAH

σ<

πAH

σ. Condition (32) then requires

πAL

σ>

πAL

σ, which impliesπA

L > πAL . However,

for all points belowA2, we haveπAL < πA

L . Therefore, at such a point belowA2, V AL is too

high, and has to be reduced.

Assume now, that condition (33) is satisfied for a contractA4 aboveA3, see Figure 8.53 AtA4, ∆mA > 0 and∆V A

L < 0. UsingV AL = V A

H − (pH − pL)v(mA), we have

∆V AL = ∆V A

H − (pH − pL)v′(mA)∆mA, (34)

for somemA ∈ [mA,mA +∆mA]. SincePAH can be rewritten as

PAH =

1

nA + nBeV BH

−V AH

σ

, (35)

for PAH to be identical for both levels ofσ, we have to have

V BH − V A

H

σ=

V BH − V A

H

σ=

V BH +∆V B

H − (V AH +∆V A

H )

σ +∆σ. (36)

Solving for∆V AH yields

∆V AH = ∆V B

H − (V BH − V A

H )∆σ

σ. (37)

Using condition (35),V BH

−V AH

σcan be expressed in terms ofPA

H as,

V BH − V A

H

σ= ln

(1

nBPAH

−nA

nB

). (38)

Substituting in condition (37) yields

∆V AH = −

∆σ

1− PBH

− ln

(1

nBPAH

−nA

nB

)∆σ, (39)

so that for∆V AL we have

∆V AL = −

∆σ

1− PBH

− ln

(1

nBPAH

−nA

nB

)∆σ − (pH − pL)v′(mA)∆mA. (40)

Rewrite condition (32) as[λPA

L (1− PAL ) + (1− λ)PA

H (1− PAH )

](SA

L − V AL ) (41)

53Note that contractA4 has to be belowA5, the contract associated with the efficient level of care: IfbothA5 andB1 were offered, almost allL-types would chooseB1.

32

−(1− λ)PAH (1− PA

H )(pH − pL)mA − [λPAL + (1− λ)PA

H ]σ = 0.

Denote byF (σ) the LHS of (41) evaluated atσ, and likewise forF (σ). If, atA4, F (σ) > 0,πAL is too large and has to be reduced, i.e.V A

L has to be increased. SinceF (σ) = 0, V AL has

to be increased ifF (σ)− F (σ) > 0. This difference is given by

[λPA

L (1− PAL ) + (1− λ)PA

H (1− PAH )

](∆SA

L−∆V AL ) (42)

−(1−λ)PAH (1−PA

H )(pH −pL)∆mA− [λPAL +(1−λ)PA

H ]∆σ = 0,

with∆SA

L = pL[v′(mA)− 1]∆mA, (43)

wheremA is defined as above. Substituting (40) and (43) in (42), we have

[λPA

L (1− PAL ) + (1− λ)PA

H (1− PAH )

] [pL(v′(mA)− 1)∆mA +

∆σ

1− PBH

(44)

+ ln

(1

nBPAH

−nA

nB

)∆σ + (pH − pL)v′(mA)∆mA

]

−(1−λ)PAH (1−PA

H )(pH −pL)∆mA− [λPAL +(1−λ)PA

H ]∆σ.

Sincev′ > 1, expression (44) is larger than

[λPA

L (1− PAL ) + (1− λ)PA

H (1− PAH )

] [ ∆σ

1− PBH

+ ln

(1

nBPAH

−nA

nB

)∆σ

](45)

−[λPAL + (1− λ)PA

H ]∆σ.

SolvingnAPAH + nBPB

H = 1 for PBH and substituting in (45), expression (45) is positive if

[λPA

L (1− PAL ) + (1− λ)PA

H (1− PAH )

] [1 +

(1−

1

nB+

nA

nBPAH

)ln

(1

nBPAH

−nA

nB

)]

−[λPA

L + (1− λ)PAH

](1−

1

nB+

nA

nBPAH

)> 0.

As can be shown numerically, this condition is always satisfied for any values ofPAH , PA

L ,λ, nA andnB as long asPA

H < 0.6PBH andλ > 0.08. Unless the share ofL-types is very

low, this condition is therefore satisfied for all reasonable values ofPAH .

If there exists a pointA4 aboveA3, so that (33) is satisfied, condition (32) is violated in away, so thatV A

L has to be increased. Therefore, the crossing of the two curves mA(V AL )

andV AL (mA) occurs to the right of the new iso-PA

H -curve, soPAH (σ) > PA

H (σ).

33

A.2 Proof that nA increases inσ

In this section it is shown that ifσ increases, the difference in profitsπA − πB at somepoint becomes large enough, so that it is profitable for a type-B insurer to become a type-A

insurer. To do so, it is shown thatπB

σdecreases (with a lower bound of zero), whileπA

σdoes

not fall below the level whenPAH = 0.

For πB

σwe have

πB

σ= (1− λ)PB

H

πBH

σ. (46)

Solving the FOC

(1− λ)PBH

[(1− PB

H )πBH

σ− 1

]= 0

forπBH

σand substituting in (46), we have

πB

σ= (1− λ)

PBH

1− PBH

, (47)

so πB

σdecreases asPB

H decreases, with a lower bound of zero, i.e.

πB

σ

∣∣∣∣PBH→0

→ 0. (48)

For insurerA, usingπAH = πA

L − (pH − pL)mA, we have

πA

σ=

[λPA

L + (1− λ)PAH

] πAL

σ− (1− λ)PA

H (pH − pL)mA

σ. (49)

Solving

∂πA

∂V AL

=

[λPAL (1− PA

L )

σ+ (1− λ)

PAH (1− PA

H )

σ

]πAL (50)

−[λPAL + (1− λ)PA

H ]− (1− λ)PAH (1 − PA

H )

σ(pH − pL)mA

forπAL

σ, and substituting in (49) yields

πA

σ=

(λPAL + (1− λ)PA

H )2

λPAL (1−PA

L ) + (1−λ)PAH (1−PA

H )+

(1−λ)λ(pH−pL)PAL PA

H (PAL −PA

H )

λPAL (1−PA

L ) + (1−λ)PAH (1−PA

H )

mA

σ.

(51)

This expression has to be compared withπA

σfor PA

H → 0, (i.e. forσ → 0), where

πA

σ

∣∣∣∣PAH→0

→ λPAH

1− PAH

. (52)

34

Note that in this case we haveπA

σ→ λ nA

1−nA , wherenA andnB is set so thatπA = πB.

It is straightforward to show that for the first fraction of (51),

(λPAL + (1− λ)PA

H )2

λPAL (1− PA

L ) + (1− λ)PAH (1− PA

H )> λ

PAL

1− PAL

. (53)

The second fraction of (51) is positive sincePAL > PA

H . It can be concluded thatπA

σ

∣∣∣PAH>0

is bounded from below byλ PAH

1−PAH

> 0, see (52), whileπB

σdecreases inPB

H , approaching

zero asPBH → 0, see (48). Therefore, ifPB

H is small enough,πA − πB is large enough, sothat it is profitable for one of the type-B insurers to become a type-A insurer.

A.3 Comparison of the pooling equilibrium and the ‘Wilson’-contract

Solving condition (22)

RA − pmA =nσ

n− 1(54)

for RA and substituting inπAH yields

πAH = RA − pHmA =

nσ

n− 1− (pH − p)mA. (55)

Substituting the condition for the ‘Wilson’-contract,v′(mWI) = p

pL, in (23), we have

[1−

λ(1− λ)(pH − pL)2

nσn−1p

mA

]p

pL= 1. (56)

Solving formA,

mA =(p − pL) nσ

n−1

λ(1− λ)(pH − pL)2, (57)

and substituting in (55) then yieldsπAH = 0. Therefore the pooling equilibrium coincides

with the ‘Wilson’-contract forπAH = 0.

35

A.4 Example with risk adjustment

Table 4: Example III withv(m) = ln(m), pL = 0.2, pH = 1, λ = 0.5, n = 20 and riskadjustment.

∑SH denotes the sum of expected surplus for theH-types,

∑SL the sum of

expected surplus for theL-types, with welfareW the weighted average of these two sums:W = λ

∑SL + (1 − λ)

∑SH . For the pooling equilibrium, all insurers are denoted as

being of typeA.

RA nA nB mA mB PAL PA

H PBL PB

H

∑PAL

∑PAH

∑SH

∑SL W

.00 12 8 .328 .981 .0831 .0157 .00036 .1014 .997 .189 -1.0838 -.2883 -.6860

.01 12 8 .340 .978 .0831 .0170 .00042 .0995 .997 .204 -1.0856 -.2836 -.6846

.02 12 8 .352 .975 .0830 .0183 .00050 .0975 .996 .220 -1.0874 -.2788 -.6831

.03 12 8 .365 .970 .0829 .0198 .00059 .0952 .995 .238 -1.0890 -.2741 -.6815

.04 13 7 .375 .971 .0766 .0217 .00063 .1025 .996 .283 -1.1009 -.2709 -.6859

.05 13 7 .390 .965 .0765 .0236 .00076 .0990 .995 .307 -1.1022 -.2660 -.6841

.06 13 7 .406 .958 .0764 .0255 .00093 .0954 .993 .332 -1.1028 -.2612 -.6820

.07 14 6 .420 .957 .0710 .0286 .00102 .0998 .994 .401 -1.1158 -.2571 -.6865

.08 14 6 .438 .946 .0709 .0307 .00128 .0950 .992 .430 -1.1144 -.2524 -.6834

.09 15 5 .455 .942 .0662 .0344 .00146 .0968 .993 .516 -1.1257 -.2480 -.6869

.10 16 4 .475 .935 .0621 .0383 .00172 .0969 .993 .613 -1.1352 -.2436 -.6894

.11 17 3 .496 .925 .0585 .0420 .00209 .0954 .994 .714 -1.1416 -.2392 -.6904

.12 18 2 .517 .909 .0553 .0452 .00265 .0928 .995 .814 -1.1445 -.2351 -.6898

.13 pooling .542 .542 .0500 .0500 - - 1.00 1.00 -1.1545 -.2309 -.6927

.14 pooling .559 .559 .0500 .0500 - - 1.00 1.00 -1.1407 -.2281 -.6844

.15 pooling .576 .576 .0500 .0500 - - 1.00 1.00 -1.1277 -.2255 -.6766

.20 pooling .661 .661 .0500 .0500 - - 1.00 1.00 -1.0751 -.2150 -.6451

.25 pooling .746 .746 .0500 .0500 - - 1.00 1.00 -1.0392 -.2078 -.6235

.30 pooling .830 .830 .0500 .0500 - - 1.00 1.00 -1.0163 -.2033 -.6098

.35 pooling .915 .915 .0500 .0500 - - 1.00 1.00 -1.0038 -.2008 -.6023

.40 pooling 1.00 1.00 .0500 .0500 - - 1.00 1.00 -1.0000 -.2000 -.6000

36

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